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The paper proposes a conversion of plane shooting targets using two dimensional affine transformations. It is assumed that affine transformations will be useful for target generation in firearms training simulators (FATS). The structure and the matrix formation algorithm of target vertex matrices are described as well as transformation matrices and output matrices. Vertex matrices were formed for twelve plane shooting targets. Transformation matrices 3×3 in size were used for conversions. Mathematical modelling and computer simulation of software-configurable plane shooting targets for FATS were performed in the Mathcad v.15 software. [ABSTRACT FROM AUTHOR]

The goal of this contribution is to reveal the analytical framework and synthesis guidelines for frontal-attacking targets (FRATs) and flankattacking targets (FLATs) from the point of view of a graph as a mathematical object. The final outcome of this study are three graph models that in many ways describe the shooting targets under consideration. The first graph model characterizes the structure of connections between the vertices using an undirected graph. The model showed that the complexity of silhouettes leads to an increase of the path in the graph and growth of the complexity of its internal structure. The second graph model allows the analysis of the connectivity of the graph vertices. In this case, a bipartite graph is used. As a result, the reviewed FRATs and FLATs are described by the same graph. The second model showed its indifference to the types of the used graphic primitives (GPs). The third graph model was developed for the analysis of the common borders of the neighboring GPs and it uses a bipartite graph. It is also indifferent to the types of the used GPs, but it takes into account the length of the common borders. The third model describes FRATs/FLATs groups in the same way. When using I-III models, one can design GPs and carry out the synthesis of new targets. A full group of flank-attacking targets consisting of five silhouettes and their GPs is offered. [ABSTRACT FROM AUTHOR]

A comparison of two groups of plane shooting targets (PSTs) characterizing the shooter-silhouette in different observation and rifle-firing positions was the starting point for this study. Selected infantry targets are used for shooting training in the Russian Federation and in the Swiss Confederation. The comparison results of two target groups (five PSTs in each of them) showed a significant similarity of their geometric shapes. To explain this fact, a targets design system (TDS) was developed. The TDS is based on attributing a certain number of simple geometric shapes - geometric primitives (GP). In our case, the number of GPs was equal to ten (five polygons for Russian and five polygons for Swiss targets). The TDS enabled building a human-like target silhouette. If two sides of two adjoining GPs or their parts become common for them, then such GPs can be combined into one common geometric figure whose area is equal to the sum of the two GPs. The TDS was further transformed into two isomorphic graphs. Their adjacency matrix (AM) was obtained. The AM matrices for the Russian PSTs and the Swiss PSTs were the same. To improve the estimation of the area and the coordinates of the target centroid, a matrix modification of Bourke’s formulas were proposed. The geometric areas for the Russian and Swiss PSTs and the location of their centroids were refined and compared. GNU Octave, GeoGebra and Mathcad were used as mathematical software for computer calculations and for graphic visualizations. [ABSTRACT FROM AUTHOR]

This article presents a method for assessing the hit probability for stationary shooting targets as a function of the projectile horizontal range (PHR), the target type and the point of aim (POA) location. The proposed procedure consists of three blocks. Block I is devoted to the double integral formula taking into account changes in the point of impact (POI) location as a function of the PHR. The characteristics of bullet accuracy and precision versus the PHR are determined in block II. The basis of blocks I and II is the bi-variate uncorrelated Laplace-Gauss probability distribution. The functions of the POI location (ordinate/abscissa) versus the PHR and the functions of the characteristics of the bullet dispersion versus the PHR are represented in the form of polynomials. The description of the target silhouette contour is given in block III. Mathematically, the target contour is a piecewise function which defines the upper and lower edges of the shooting target and it also represents the limit for the double integral formula of block I. The proposed method is built on a modular basis and allows a user to change types of weapons and shooting targets. For demo calculations, the accuracy and precision characteristics of the 5.45×39 Kalashnikov assault rifle model MPi AK-74N were selected. Five types of Swiss military targets imitating an OPFOR combatant's silhouette were used as shooting targets. For illustrating the operability of the proposed method, the hit probabilities for the K, H, G, F, and E shooting targets were evaluated for the PHR from 50 to 400 m. All computations are implemented in the PTC Mathcad v.15. [ABSTRACT FROM AUTHOR]

geometric interpretation of single shot hit probability (Phit) is a volume of the 3D space under the surface f(y,z) described by the bivariate normal distribution and bounded from below by the YOZ plane with the target’s contour (T-region). The Phit is proposed to be estimated by a method based on the numerical integration of the double integral. The double integral integrand is the 2D normal distribution of the Y, Z system of random variables. The dispersion characteristics and the coordinates of the dispersion center are known in advance.The limits of the first and the second integral are described by the analytic functions which characterize the geometric shape of the T-region. The implementation of the offered method is as follows: the selected shooting target is partitioned into N geometric subregions and then analytic formula(s) for each subregion’s boundaries is/are determined and each double integral is defined. The Phit estimations are produced using a numerical integration in the computer software Mathcad. The results of the calculus of all Phit values (for subregions) are added up (or subtracted) depending on the geometric relationships between the regions. The schema for solving Phit numerically makes it possible to calculate the likelihood for targets with arbitrary geometric shapes and not just for rectangular-shaped silhouettes. For illustrating the operability of the proposed method, the Phit for two kinds of head-type shooting targets has been evaluated. The developed method has been compared with the already existing works. [ABSTRACT FROM AUTHOR]

A description of a pistol (rifle) cartridge often involves two ballistic coefficients that characterize its ballistic qualities with respect to various air resistance laws (ARLs). How close are the obtained ballistic trajectories with varied ARL specifications and what are the differences between them? How to evaluate ballistics if the ARLs are to be expressed in various mathematical forms? In this paper, the evaluation of external ballistics trajectories is given for two ARLs (the law brought in 1943 and the Siacci law). All the obtained results relate to the TT pistol with 7.62 × 25mm Tokarev cartridge.The paper also presents the answer to the question: how to calculate the ballistic trajectory if the ARL is expressed as a rational function, piecewise function or spline. For the 1943 ARL, a graphical interpretation of the function Cd (i, v) in the form of a surface is shown. This paper shows that, due to the selection of ballistic coefficients, it is possible to obtain sufficiently similar form of ballistic trajectories. A method of graphical comparison of external ballistic parameters is presented as well as the mathematical tools for quantitative analysis of a shape of ballistic curves.The difference between the two trajectories is proposed to be estimated using a relative error in regard to а selected ballistic parameter. Computer simulation considered for the 1943 and Siacci ARLs for the 7.62×25mm Tokarev cartridge indicates that the profiles of the function of instantaneous projectile velocity vs time of flight (TOF) had the greatest non-coincidence in relation to other ballistic parameters (e.g. horizontal range, height of the trajectory, etc.) The obtained maximum of the relative error was 0.8%. Its magnitude localizes at the point of impact. [ABSTRACT FROM AUTHOR]

The paper presents mathematical modeling and computer simulation (MM&CS) in the area of numerical solving of the basic problem of external ballistics for tubed artillery. Five-stage MM&CS scheme for conducting a ballistic simulation is developed. It is shown that a formal mathematical procedure allowing to solve the basic problem of external ballistics is a numerical solution of the Cauchy problem for a system of ballistic differential equations. The trajectory of a projectile flight for the 57-mm ZIS-2 anti-tank cannon is estimated. A solving algorithm and the Mathcad program code are given.The numerical solution for a system of four first order ballistic differential equations is a five-dimensional space.The possibility of visual presentation for a numerical solution was proposed in the form of a square matrix. The boundaries of each subspace are determined. A procedure based on spline functions is developed for checking the correctness of the numerical solution. As a result of such verification, the effects of a light increase in the error at the edges of the integration interval are observed. A comparison of the numerical solution of the basic ballistics problem is conducted by means of “soft“ and “stiff“ solver-functions.The trajectory parameters estimated by “soft“ and “stiff“ methods are the same up to the fifth decimal place. [ABSTRACT FROM AUTHOR]

The estimating of a projectile initial velocity is formulated as a two-point boundary value problem. To solve it, the data of a Doppler Radar or the results of solving the Cauchy problem can be used. The projectile initial velocity v0 estimation process is based on the numerical solution of a system of ordinary differential equations and the bisection method. The iterative calculating process is interrupted when a predetermined accuracy of a projectile initial velocity and a predetermined value of the width of velocity's search interval is reached. In the article, the block diagram of the algorithm for the projectile initial velocity process is developed. The Mathcad program code for mathematical modeling and a computer simulation of the projectile initial velocity estimation process for a 57mm armor-piercing projectile of ZIS-2 anti-tank gun 1943 model is given. [ABSTRACT FROM AUTHOR]